-*---* A. Iarrobino, Math U141, Fall 07 Classnotes

Math U141 class notes, Fall 2007

Prof. A. Iarrobino


For Sections


meeting MWTh at 9:15AM in 178FWV and 10:30 AM in 140 Dodge.


Look here for specific assignments and brief comments on what we did (or will do) in class. This will be updated at least weekly and usually by 4:30 PM each class day. TBA = "to be announced"


Course-wide Spring 07 syllabus info (Prof. A. Iarrobino coordinator): Math U141 Course wide info (Syllabus A) (pdf)

Information sheet (grading, etc. for Prof. Iarrobino's sections) Math U141 Information Syllabus B (pdf file)

Course home page (you probably already know this information): Math U141 home page (html)

Course outline (broad strokes) Course Outline



First day of classes: Wednesday September 5 Go over syllabus, information, take placement quiz (for student info only). Review the equations of a line, both slope-intercept y=mx+b form and point slope form. Finding points a fraction - as two thirds - along the line segment between two points.

HW for Thursday September 6: Worksheet #1A (handed out in class), #1A-E.

Thursday, September 6 Placement quiz back, we went over it.
We went over WS#1A, problem #1. We emphasized the role of units in equations: using #balloons, minutes vs. #balloons, hours led to different equations.
Then we studied average rate and instantaneous rate of change for y=5x^2 (giving height of ballon in feet x minutes after start) between x=3,x=5 (obtaining 40 ft/min) then between x=3, and x=3+h, average rate of change
m_{PQ}= 30+5h.
We then discussed the limits as h goes to zero, of m_{PQ}, obtaining
m_P=lim_{h to 0} m_{PQ}=30 ft/min,
the instantaneous rate of change of height with respect to time at time 3 minutes.

Note for more information on limits, see p. 135-139, especially examples with quadratic functions (not exponential yet).

HW for Monday: Worksheet #1A, #2.3
Also Text Section 1.2 p. 11 #2,7,9,15,21,24,26-29.
Text Section 2.1 p. 103 #1,2,3.

Week of September 10 Continue discussion of average, instantaneous rate of change. Derivative function. Quiz #1 Thursday.

Monday September 10 (Plan) Check HW. Go over HW.
Continue discussion of average, instantaneous slope.
Introduced derivative function, Interpretation of derivative function as slope of f(x) at any point.
Finding the equation of tangent line at a particular point. Example, y=3x^2, we found the slope function y'=6x.
Thus when x=-5, slope f'(5)=6(-5)=-30, the eqn of TL through the point P(-5,75) is : y-75=-30(x+5).

HW for Wednesday: Quiz 1 Spring 06, passed out and on web.

Next Office Hours Wed. Sept 12, 12:30-3, at 526 NIghtingale.

Tutoring (Math dept, free) 540B Nightingale: Wed Sept 12, 1-4 PM.
Full tutoring hours begin September 17, and will be M-W 10 AM-9 P, Thurs 10-6, Fri 10-1.

Wednesday Sept 12: (plan) Slope function for polynomials. Application of derivative to motion. Concavity.

Thursday Sept 13 Questions, derivative, Quiz 1.

Homework for Monday Sept 17 Read text section 2.2. Do p. 109 #1-13,15,19-26,29.
Do Section 2.3 p.116 #1,2.

Week of September 17: Derivative functions (2.2), interpreting the derivative (2.3), Using the derivative to find max/min (intro) WS 2B, second derivative (2.4), intro to limits (p.135 ff), Derivative formula for powers (class and (3,1)).

Monday September 17 Quiz 1 back with solutions. Questions on HW. #30 section 2.2 (9:15 class only)
WS 2B #1 (pollywog problem).

HW for Wednesday
Quiz 2a p.38 of packet #3 (like pollywog problem done in class)
Text: syllabus HW Section 2.3: p.116 #1-2,5-7,9-13,15,18,23,27-30,31-35,37.

Wednesday Sept 19: Interpreting derivativem linear approximation
Derivatives of powers

We began with Section 2.3 #12,27,29,37: used the formula
Delta y = f'(x) Delta x (text p. 114 or 115).
Then we derived the formula for derivative of f(x) = x^n, f'(x)=nx^{n-1} (see section 3.1), using the binomial theorem.

m_{PQ}=[(x+h)^n-x^n]/h = nx^{n-1}+ h(further terms)

Then m_P= lim_{h \to 0} m_{PQ}= nx^{n=1}.

To use this formula on a function f(x) involving roots and powers of x, first convert f(x) to a standard sum of (fractional, if needed) powers of x, then apply the derivative formula.
Note that the deriative of a constant function is 0.

Homework for Wednesday Section 3.1 # 5-7, 9-19
Worksheed 2B #3A,B (only)
Read Section 2.4 (second derivative). (Optional: Do #1-7).

Thursday September 20
Questions on power rule: Volunteers did WS #2B #3B, Text chap 3.1 #10,19 on board.
Derivative formula for exponents, ln, and sinx, cosx; distinction between exponential function e^x and power function x^2. x^n (power a constant).
Use of derivative function to
-- find slopes, then equation of Tangent Line at a point on a curve y=f(x) (WS 3A #2A,D done in class).
-- find local max-min: y=x^3-6x^2+2 done in class. derivative y'=3x^2-12x, is 0 at x=0, and at x=4. We get points P(0,2), Q(4,-30) on curve where the slope is zero. Since y''=6x-12 is negative at x=0, the curve y=f(x) is concave down at P, and P is a local max.
Since y''(4)=12 is positive the curve y=f(x) is concave up at Q, which is a local min of curve.
We used graphing calculator to compare exonential and power functions, and for y=x^3-6x^2+2.

Meaning of derivative function (review) the derivative function gives the slope of the original functionat above each x-value.
For y=x^3-6x^2+2, we noted that the value y'(2)=-12, is the slope of y=f(x) at the point (2,-14).
Second derivative: we briefly discussed it for graphing (see above). We also discussed #9 section 2.4.

HW for Monday Sept 24
Worksheet #3A, p.11 of packet.(Note, there are no answers to WS in packet).
Text section 3.1 #37,39,40,4345,51,59.
Section 3.2 odd #1-9
Begin (so optional for Monday) 2.4 # 9,11-12,15-16.

Week of September 24 Continue derivative formulas, exponentials and composite functions, chain rule. Second derivatives. Application to graphing. Quiz 2 on Thursday.

Monday September 24 Questions: WS 3A #3, Section 3.2 #44,45,59.
Local linear approximation (p. 115 of text, or use tangent line).
Composite function (Section 1.8)
Chain rule, extended power rule (Section 3.3).

HW for Wednesday Sept 26:
Section 3.3 p.157 #1-17 odd.
WS 3B #5A-C (only)
Quiz 2a, p. 37 in packet do #2-4 (preparation for Quiz 2).
[Added after class. so optional for Wed.] Read Section 1.8, be able to do p. 55 #9,11,#36 write a table for f(g(x)) only.

Wednesday September 26 Questions: We went over WS 3B #5B, Quiz 2a #4D, and several text Section 3.3 problems (#7,17), also (10:30 AM class) #35 p. 157.
Exponential functions: equal ratio in equal time, formulas f= A_0 b^t, b is ratio, b=1+r, r = rate (see worksheet #5).
Second formula f=A_0 e^{kt} , k=ln (1+r).
Product rule, begun from WS 3B #4 (see also Section 3.4).

Homework for Thursday Sept 27
Text Section 1.5 p. 38 #1-4, 6, 9,11,13,17,25-26.
WS 3B #4A-C.6B (9:15 AM - begin #7)
Section 2,4 #1-7 (second derivative, concavity)
(Optional) begin Section 3.4 HW for Monday
Note: Quiz 2 preparation: Quiz 2a #2-4, also Exam 1a p. 45 #2, 3A,3B, 4. (Note: local linear approximation is included on the quiz).

Thursday September 27 (Plan) Questions, Product rule, Quiz #2 (40 min).

Homework for Monday Oct 1
(Finish HW for Thursday, see above)
Section 3.4 p. 161 #2, 3-15 odd, 34,41

Week of October 1 Continue Product and quotient rule (Section 3.4), begin trig functions (Section 1.10) and their derivatives (section 3.5), graphing using calculus (Section 4.1)

Monday Oct 1 Discussion of quiz 1. WS 3B #7, product and quotient rule, more complex derivatives.

HW for Wed. Oct 3
9:15 AM class: WS 3B #6. Finish product rule HW section 3.4.
10:30 AM class: Section 3.4 #1-39 odd.

Wednesday Oct.3 Questions on HW (9:15 #5D,E, #6A,B,C on Worksheet 3B; some Section 3.4); 10:30 various in Section 3.4)
Trig functions: amplitude, period,displacement and y=Asin(Bx)+C. Modeling data with a trig function: Section 1.10 #1,31 in class.

HW for Thursday Oct 4
Section 1.10 p. 68 #1-5, 13,31,34 (note #1,31 done in class).
Section 3.5 p.165 #1-ll odd
Packet Quiz 3B #6 (if you haven't already completed them).
9:15 class: Class Packet Exam 1a p.45,47 #3 and #7 (complex derivatives)

Thursday October 4 (Plan) Questions on trig functions, complex derivatives.
Local max, min, inflection points, graphing using calculus (Sections 4.1, 4.2).

Howework for Wednesday Oct 10
Section 3.5 p. 165 #13-19 odd, 22,23,26.
WS 3B #6 (if not done), Exam 1a p. 45 #2,3,7 (if not already done)
Section 4.1 p. 180 #1-5,8-10,14-15,21,27,28.
Section 4.2 p. 186 (begin) #1-6,8-10, 11-15 odd, 26, 27-31, 33.

Week of October 8 Graphing using calculus, global max-min

Monday Oct 8 No class: Columbus Day, NU closed.

Wednesday October 10 Graphing continued, inflection points.
Global Max-min (section 4.3).

Homework for Thursday Oct 11 Complete Section 4.2 HW.
Section 4.3 p. 191 #1-3,5-9,11-20,28-30.

Thursday October 11 Questions, global max-min.
Half-quiz.

Homework for Monday October 15:
Exam 1a in packet
Finish section 4.1-4.3 homework.
Read p. 185 text, Example 5. Do p. 187 #27-31.

Week of October 15 Practice for Exam 1, begin max-min work problems.

Monday October 15 Half quiz 2.5 back, work in groups to correct it; Problems in class.
Max-min work problems (begin).

Homework for Wednesday October 17
Do Exam 1 from Fall 06.
Exam 1, Math U141 Fall 2006

Wednesday October 17 Practice for Exam 1

Thursday October 18 Exam 1.

Homework for Monday Oct 22 Prepare for max-min word problems, which we'll begin on Monday.
Read in Class Packet p. 15-20.
p. 15 WS 4C, try #2 p. 15 using same method.
p. 17 read #1-4.
p. 19, 20 (work in Math U141, grading).

Exam 1, Math U141 Fall 2007 Solutions (pdf)

Week of October 22 Max-Min word problems. Logistic growth, Surge function. Possibly begin antiderivatives.

Monday October 22 Max-min word problems, strategy in two variable problems as minimizing lengths of fences, or maximizing area of garden. Other types.

Homework for Wednesday October 24.
Read class packet, p. 15-20.
Do p. 21 (header MM #1): #1-6, 8-9. If you feel comfortable with area formula A= 0.5 ab sin theta for a triangle, you could try also #7.

Wednesday October 24. Checked HW. Max-min word problems, continued. Students put on board p. 21 #5,8, we went over #7.
We discussed problems using a linear functon: as peach orchard, timing of orange picking: p. 22 #18,20,21.

Homework for Thursday, October 25. p. 21-22 #10,15-21.
Note: #15: use Volume = lwh, and express w,h in terms of x, the side of the corner cut out.
Note #16,17: Use that profit = $ sold - $ cost. The fixed cost is for setting up (so is not multiplied by number of units sold). You could read section 4.4 for a slightly different approach (see Example 3,4 p. 198-199 for the kind of problem done in class. This section uses a notation of ``marginal cost'', ``marginal revenue'', it may be simpler to omit (not assigned).

Thursday October 25 Check HW. Max-min continued: students did problems p. 21-22 #10,21 in class; we also did #15, 16, and set up #24 p. 22.
Antiderivatives (section 7.1).

HW for Monday Oct 29. Class Pack p. 21-22 #22,25.
Text: Read Section 7.1. Do p. 303 #1-21 odd, #41-47 odd.

Week of October 29 Antiderivative, Motion problems, further max-min, logistic, surge functions. Quiz 3 Thursday.

Monday October 29 Questions on antiderivatives, max-min word problems
Antiderivative of u^n where u is ax+b; also, the antiderivative of (u^n times u') is u^{n+1} /(n+1)+C.
Motion word problems using antiderivatives: method obtaining position, velocity, acceleration, from each other using derivative, antiderivative, and initial conditions.
WS 4A #1, 4A #2 started both classes
WS 4B #1 in 9:15 AM class.

HW for Wednesday October 31.
Class Packet p. 24 AA, #1-13
WS 4A #2, try 3; WS #4B, #1,2.

Wednesday October 31 Questions on motion problems: #2,3 on WS #14A (p. 13), #2 on WS #4B.
Motion, continued: #1,5 p. 25 of packet.

HW for Thursday, November 1. P. 25-26 of packet, #1-12.
Prepare for quiz 3 (especially 9:15 AM class).

Thursday Nov. 1 Questions
9:15 AM: Quiz #3.
10:30 AM: practice for Quiz 3#, logistic equation.

Homework for Monday Nov. 5.
Complete HW above for Thursday.
Read Section 4.8 (surge function), try problem #1.
Read section 5.1, do problem #1.

Week of November 5. Surge function (application of max-min to medicine absorption)
Integration

Monday November 5. Questions. Check HW.
Surge function, integration (9:15 AM);
Surge function, and quiz 3 (10:30 AM).

Homework for Wednesday Nov. 7
Section 4.8 surge function p. 225 #1-4,6,9.
9:15 AM: also class packet p. 29, WS #6A #1,2

Wednesday, November 7 Definite integral as area
Section 5.2: Left, right and trapezoid sums for the definite integral.

Homework for Thursday, Nov. 8
Section 5.2 p. 247 #1-3,5-7.
WS # 6A p. 29 #1,2,3
WS #6B #2 (added on web).
(optional) WS #7A p. 33 #2 (we'll discuss this Thursday)

Thursday November 8 Integral, continued, interpretation of integral.
(Sections 5.3-5.4 and worksheets).

Homework for Wednesday, Nov. 14
Section 5.3 #1,3-7
Section 5.4 #11
Section 5.5 #11,13, 44

Week of November 12 Monday holiday - NU closed. Integration, continued, fundamental theorem of calculus. Text sections 5.5, 7.3.
Quiz 4 on Thursday (see below)

Wednesday November 14 Integral, definite integral, application of definite integral as change in amoount from rate of change

Homework for Thursday, November 15
Section 5.4 #23-25, 30, 31
Section 5.5 #11,13
Section 7.3 #1-6
Class pack p. 71 #11 (from final exam F2003)
Prepare for quiz 3.5: Classpack p. 41-42: Quiz 3.5, 4 (except #2).

Thursday November 15 Questions on Definite intregral from HW.
Area between two curves (Section 5.3).
Half Quiz 3.5 (30 min).

Homework for Monday November 19
Section 5.3 p. 254 #23,25,27-29, 31-32.
Packet p. 42, Quiz 4 #2
Prepare for Exam 2: Packet p. 47-49. Exam 2 Fall 03, but for #7 only #7B.

Monday November 19 Quiz 3.5 passed back with solutions. Quiz 3.5 solutions (pdf)
We went over the integration problems on Quiz 3.5.
Area between curves, setting up integral
Area using calculator.

No class Wed. Nov 21 and Thurs Nov 22 (Thanksgiving holiday).

Homework for Monday Nov. 26:
Area between curves (complete HW for Nov. 19).
Prepare for Exam 2: Exam 2 in packet, p. 47-49 (only 7B from #7)
Sample integration problems from packet p. 51 except #1D
p. 52-53 Math 1108 Exam 1 except #8.

Week of November 26 Practice for Exam 2, Exam 2, differential equations.

Monday November 26 Questions, Practice for exam 2

Wednesday Nov. 28 Exam 2

Thursday Nov. 29 Differential Equations, Section 10.1, 10.5.

Exam 2 (9:15 AM) solutions (pdf) The 10:30 AM exam 2 differs only in #2 height of Callalah, and #2D.).

Homework for Monday Dec. 3
Class Packet p. 43 Half Quiz 4 #2,3.
Text Read Section 10.1, do p. 401 #1,3,5-7,9,12,14-15.
Text Read Section 10.5 p. 421ff. Do p. 423 #12,13,16,17.

Week of December 3 Mon and Wed: Differential equations, questions, practice for final exam.

Monday December 3 We did #2,3,4B on p. 43 of packet.
9:15 AM: We also did, with help of Joanna Bakos and Kim Patch, #12. p. 425 of text.

Solution to #12: : B'= 0.07B-1000, standard form B'=-0.07(14,285-B).
Equilibrium (B constant so B'=0) of B=1000/0.07=14,286. Unstable equilibrium (see below).
Solution to y'=k(a-y) is y=a+Ce^{-kx}, so here B=14,285+Ce^{0.07x); (note k is negative, so -k in exponent is positive)
Find C: B(0)=10,000 = 14,285, C=10,000-14,285=-4,285,
Solution B=14,285-4,285 e^{0.07x). Note this goes to -infinity as x gets high, since the exponential term is e^{positive times x) so gets very large as x increases.
So the equilibrium here is unstable: non-equilibria solutions flee the equilibrium as x gets large.
Stable equilibrium: NLC temperature problems, or medical ones, where all the solutions approach the equilibrium value as x increases.

10:30 AM: Passed back exam 2, went over problems 4-6 on the exam 2.

HW for Wednesday Final Exam 2003 fall from class packet, p. 73.

Wednesday December 5 Practice for final exam. Questions

Reminder: Extra Credit HW At the final exam, pass in notebook or collected HW, with pages/sections identified.
You get an ECHW grade based on amount of HW completed, and that may help the Quiz-HW grade. I go over these and you pick them up when you leave the final exam.

Office hours during Exam week, etc:
Monday 12:30-1:30, (then I'm at seminar), then 3-3:30+
Wed. 1-2, 3:30-4+ (later if needed).

2 Review sessions:
A. Monday 4 PM, room 305 Shillman.
(Dr. Marina Ville leading, it is open to all Math U141)

B. Wednesday 2-3:30 PM 205 Shillman Hall
Dr. Ville and others (I plan to be there for the first half).
{review sessions are question driven, bring in your questions).

Extra Credit Homework (bring in to final exam), See syllabus or class notes page.

Other info re preparing for final.
There are two finals with solutions in
the class packet; and the sample questions for final exam (p. 62-66), and as well another final without solutions.
The Math Office has more (as Spring 06).
Math U141Fall 06 FE (pdf)
Math U141Fall 06 FE Short answers (pdf) by Prof. Krikorian
Note: these are just o check your answers, as always, you will need to show work in order to get credit

A good way to study is to do problems under exam conditions, then work with others.

Final exam: Part I (five questions, all counted). Part II (best 5 of 6).
Thursday Dec. 13 at 10:30 AM in 300 Richards.
(If you have a conflict, get very ill, have family emergency, please send me e-mail about it. If you are late come to the exam room anyway.)

Any questions: feel free to come to office hour, or send e-mail. I check e-mail frequently.

Forthcoming featured events

Final Exam Thursday Dec. 13 at 10:30 AM at 300 Richards. Two hour exam
Be sure to make your travel plans so you can take the final, required of all. Bring graphing calculator and Extra Credit HW (see above).
Archived events

Exam 2 Wednesday Nov. 28 (both sections). See Exam 2 in packet. p. 47-49, and also Integration problems p. 51-52.
More detail in Homework for Monday Nov. 26.

Exam 2 (9:15 AM) solutions (pdf) There is a typo: #6B ans is -240 ft/min^2 for av accel.

Quiz 3,5 Thursday Nov. 15. Integral, fundamental theorem, applications.
Quiz 3.5 solutions (pdf) Practice: See quiz 3.5 and 4, packet pages 41-42, except Q4 #2.

Quiz 3 Thursday November 1 (9:15 AM section)
Quiz 3 (9:15 AM) solutions (pdf)

Quiz 3 Monday November 5 (10:30 AM section).
For sample see Class Packet Quiz #3. Also consider other max-min problems and logistic and surge fctns as done in class.

Exam 1 Partial Make up: Wed Oct 31, 12:15-3 PM, meet at 526 NI, possible B+.
Exam 1 partial make up directions (pdf)

Group A: (90 possible) if you can meet with me Monday afternoon (3-4:30), with your old exam, and choose which questions to try, you will be able to begin the partial make up when you arrive Wednesday.
Group B. (+10 pts max); Please bring old exam with you. Then I will give you a problem(s) sheet to work on with 2 problems, with parts similar to two parts from #3,4,5, or trig functions (an alternate to a part of #3). Chosen at random. (So to prepare, you would want to study each topic).

Exam 1 Thursday October 18, or earlier that week. See Exam 1a in Class Packet for practice. See Exam 1 Fall 06 (above)
Exam 1, Math U141 Fall 2006 Solutions

Half Quiz 2.5 Thursday October 11. For practice see Quiz 2.5 in packet.

Quiz 2 Thursday September 27. (See sample p. 37 in class packet, solutions p. 75. Also, for further practice, Exam 1 p. 45 #2,3A,3B,4). The quiz will include local linear approximation (olong the tangent line).

Quiz # 1: Thursday September 13.

Class Packs available They were ready Sept 10 at Reprographics (Ell Building corridor behind - in back of - the book store, not accessible from bookstore). $5 each.

Reminder: attendance policy The attendance policy (announced in Syllabus: information sheet):
Briefly, more than four unexcused absences affects your grade, or may result in a W or F in the course.


Questions or comments: e-mail to iarrobin@neu.edu.  This is the quickest way to reach me, or come by 526 NI, MWTh.

Math U141 Fall 2006 Prof. Iarrobino section Information Information, Office hours, grading, etc for Prof. Iarrobino section

Math U141 home page (html)

Math U141 Spring 2006 Classnotes (may give an idea of timing of future material, quizzes, exams).

Math U141 Fall 2005 Classnotes (may give an idea of timing of future material, quizzes, exams).

Math U141 Fall 2006 Classnotes (may give an idea of timing of future material, quizzes, exams).

Math Tutoring  Free, at NU, available to Math U141 students.



Links  to other calculus resources on the web:

Visual Calculus (U. Tenn)


Prof. Anthony Iarrobino
Department of Mathematics
Northeastern University
Boston, MA, 02115
Office:526 NI
Phone: (617) 373-5524
Email: iarrobin@neu.edu


Created: September 6, 2007. Last modified: December 11, 2007. URL:http://www.math.neu.edu/~iarrobino/AIMathU141F07classnotes.html